Path Integral Approach In Strongly Correlated Electronic Systems

The rich variety of novel physics in strongly correlated electron systems have at-tracted long-term attention of the community,that include Mott transition,colossal magnetoresistance,fractional quantum Hall effect,high-temperature superconductiv-ity,heavy fermion,and so on.In these systems,due to the strong interaction between electrons,the traditional band theory based on single-electron approximation and the perturbation theory based on expansion over a small parameter do not work.There-fore,development of new methods to deal with various equilibrium and non-equilibrium physics in strongly correlated electron systems is an important,frontier,and fundamen-tal subject.This is also the purpose of this thesis.This thesis can be divided into two parts.In the first part,we develop a new method to study equilibrium quantum phase transitions.In the second part,we develop a new method to study non-equilibrium dynamical quantum phase transitions.Heavy-fermion systems are arguably one of the most widely studied strongly cor-related electron systems.Traditionally,the understanding of heavy-fermion physics is mainly based on the mean-field theory with a static and uniform hybridization approxi-mation.In this theory,the localized electrons undergo a localized-to-itinerant transition below a characteristic temperatureT_Kand then hybridize with conduction electrons to form a heavy electron band.The mean-field theory is simple and intuitive,but it ig-nores the fluctuations and correlations of the hybridization fields,which results in a phase transition atT_Krather than a crossover observed in experiment.To address this problem,we propose a static auxiliary field approximation to study the hybridization fluctuations in heavy-fermion systems and use the mutual information to measure in-tersite hybridization correlations with the help of recently proposed mutual informa-tion neural network algorithm.When applied to the two-impurity Kondo model,we find that the artificial phase transition predicted in the mean-field approximation is sup-pressed,indicating that our new method can indeed capture the fluctuation effects.In addition,we find a logarithmically divergent amplitude mutual information with lower-ing temperature near the so-called“Varma-Jones”fixed point,manifesting a quantum phase transition at zero temperature and its associated quantum criticality above the fixed point.In the Kondo regime,a large phase mutual information reveals the deep relationship between the intersite phase coherence of the hybridization fields and the hybridization picture in heavy-fermion systems.Our method can be easily extended to multi-impurity and lattice models,and we hope it will provide a microscopic theoreti-cal support for the two-fluid phenomenological model of heavy electron physics and the two-stage hybridization scenario observed in experiment.Due to the rapid development of experimental techniques,people can not only study equilibrium phase transitions and critical phenomena in strongly correlated elec-tron systems by tuning parameters such as temperature,magnetic field and pressure,but also observe non-equilibrium properties in these systems by tuning time.Especially with the discovery of many novel phenomena such as fluctuation relations,many-body localization,time crystal,dynamical quantum phase transition,and so on,the study of non-equilibrium quantum physics is becoming one of the most active and exciting branches of modern condensed matter physics and has attracted intensive attention in recent years.Quantum work is arguably one of the most important quantities to charac-terize the non-equilibrium dynamics,because it can not only describe the critical phe-nomena of equilibrium phase transition,but also capture the non-equilibrium dynami-cal quantum phase transition and fluctuation relations.In the study of quantum work in many-body systems,a Hamiltonian approach has often been used by calculating explic-itly time evolution of the quantum state for quench protocol.However,such calculations can be very involved which makes it hard to be extended to arbitrary time dependence and general correlated many-body systems,which greatly limits our understanding of non-equilibrium physics.To overcome this issue,we introduce the functional field in-tegral approach to study the statistics of quantum work and the dynamical quantum phase transition under non-equilibrium conditions,and derive the general formalism for a bilinear Hamiltonian with arbitrary time dependence.When applied to the trans-verse field Ising model,it yields the same results as the traditional Hamiltonian method,which demonstrates the correctness and effectiveness of our new method.When fur-ther applied to a topological model,we observe an anomalous 1/correction in the scale of irreversible work due to the topological nature of the quantum phase transition.Dynamical quantum phase transitions are observed for three different time evolution protocols but their time periodicity only appears in the double quench case.Finally,we take the Ising-Kondo model as an example to illustrate how to extend the path integral and Monte Carlo method to the study of non-equilibrium physics in strongly correlated electron systems.